Final answer:
To find the slope of the tangent to the curve y = 8 - 5x² - 2x³ at the point where x = a, we need to find the derivative of the function and then substitute the value of a into the derivative. The slope of the tangent line at x = a is -10a - 6a².
Step-by-step explanation:
The slope, or gradient, of a curve at a particular point is equal to the slope of the tangent line to the curve at that point. To find the slope of the tangent to the curve y = 8 - 5x² - 2x³ at the point where x = a, we need to find the derivative of the function and then substitute the value of a into the derivative.
- Find the derivative of the function y = 8 - 5x² - 2x³. The derivative is dy/dx = -10x - 6x². (Note: The derivative gives us the slope of the tangent line at any point on the curve.)
- Substitute the value of a into the derivative. The slope of the tangent line at x = a is -10a - 6a².
Therefore, the slope of the tangent to the curve y = 8 - 5x² - 2x³ at the point where x = a is -10a - 6a².